pr.probabilidad – Conditioning in an irrelevant variable in a martingale control problem

Suppose I have two independent Brownian movements. $ B ^ 1_t, B ^ 2_t $ Y $ mathbb F_t $ Be the natural filtration generated by them. Leave $ T> 0 $ be a fixed finite number Leave $ q_t $ be a $[-1,1]$ valued $ mathbb {F} _t $ Martingale that controls the analyst.

Leave $ mathcal Q $ be the set of $[-1,1]$ valued $ mathbb F_t $ Martingales The control problem is: $$ sup_ {q in mathcal Q} E_ {0, q_0} f (q_T, B ^ 1_T) $$

where $ q_0 $ is the value of $ q $ at the time $ 0 $. $ f ( cdot, cdot) $ It is linear in each argument. As we can see, the objective function does not depend on $ B ^ 2_t $.

I would guess that, given the linearity of $ f ( cdot, cdot) $ in each of its arguments and without dependence on $ B ^ 2_t $, that we can restrict attention to $[-1,1]$ valued $ sigma (B ^ 1_t) $ where the martingales $ sigma (B ^ 1_t) $ It is a coarser filtering generated only by $ B ^ 1_t $.

Does that sound like a reasonable guess? And how can I argue this if it's true?